Isotropy

The transversely bianisotropic uniaxial medium considered in the present article can be obtained, e.g., by mixing metal helices with an isotropic base medium in such a way that the axes of the helices are randomly oriented but perpendicular to a fixed direction in space. The medium is a generalization of the well-studied chiral medium, and is somewhat similar to the recently studied axially bianisotropic uniaxial medium, which has interesting polarization properties. Plane-wave propagation in the medium is studied and the solution for the dispersion equation is given. Numerical examples for the wave numbers corresponding to the two eigenwaves of the medium are displayed. It is seen that, unlike in the axially bianisotropic uniaxial medium, there are no optical axes in the present medium, in general. © 1993 John Wiley & Sons, Inc.

Shales are a major component of sedimentary basins, and they play a decisive role in fluid flow and seismic-wave propagation because of their low permeability and anisotropic microstructure. Shale anisotropy needs to be quantified to obtain reliable information on reservoir fluid, lithology and pore pressure from seismic data, and to understand time-to-depth conversion errors and non-hyperbolic moveout. A single anisotropy parameter, Thomsen's δ parameter, is sufficient to explain the difference between the small-offset normal-moveout velocity and vertical velocity, and to interpret the small-offset AVO response.

Ingenieurwerkstoffe neigen bei erh€ o ohten Temperaturen zus€ a atzlich zum elastischen Verhalten zum Kriechen oder, mit anderen Worten, man kann neben einer spontanen Werkstoffreaktion auch eine zeitabh€ a angige beobachten. Unter Ber€ u ucksichtigung der wachsenden Sicherheitsanforderungen f€ u ur Kraftwerke, Flugzeugkomponenten, chemischen Anlagen usw. ist das zeitabh€ a angige Werkstoffverhalten in den Konstruktionsprozess einzubeziehen. Zahlreiche Bauteile, die unter Kriechbedingungen eingesetzt werden, k€ o onnen als d€ u unnwandig klassifiziert werden. Ihre Analyse ist dann mit drei Problemen verbunden: mit der Auswahl eines geeigneten Werkstoffmodells, mit der Auswahl eines ad€ a aquaten strukturmechanischen Modells und mit der Auswahl eines geeigneten numerischen L€ o osungsverfahrens. Dabei besteht eine enge Verbindung zwischen den drei Problemen. Beispielsweise hat die Auswahl des strukturmechanischen Modells (Balken, Platte, Schale usw.) signifikanten Einfluss auf den numerischen Aufwand. Andererseits wird die Berechnungsgenauigkeit durch das Materialverhaltensmodell beeinflusst, wenn verschiedene Effekte einbezogen oder vernachl€ a assigt werden. Die Kriechmechanik als Zweig der Festk€ o orpermechanik hat eine mehr als einhundertj€ a ahrige Geschichte. Nach einer kurzen Diskussion der geschichtlichen Entwicklung sowie der Anf€ u uhrung ausgew€ a ahlter Referenzliteratur werden zwei Modelle zur Beschreibung des Kriechverhaltens eingef€ u uhrt. Das erste Modell ist dadurch gekennzeichnet, dass die Art des Spannungszustandes keine Rolle bei der Beschreibung des Materialverhaltens spielt, w€ a ahrend in die Formulierung des zweiten Modells eine derartige Abh€ a angigkeit eingeht. Ein typisches Beispiel f€ u ur das zweite Modell ist ein unterschiedliches Zugund Druckverhalten. Dieses kann beispielsweise im Terti€ a arkriechbereich vorausgesetzt werden, da eine Sch€ a adigung durch Zugbedingungen gef€ o ordert wird. Wenn man Druckbedingungen antrifft, k€ o onnen Kriechverzerrungen auftreten, der Sch€ a adigungszustand ist jedoch teilweise eingefroren (u.a. kein Auftreten neuer Hohlr€ a aume). Der letzte Teil ist strukturmechanischen Modellen gewidmet. F€ u ur d€ u unnwandige Strukturen werden in der Regel alle Annahmen in € U Ubereinstimmung mit der ,,D€ u unne'' der Struktur getroffen. Damit kann man beispielsweise Hypothesen f€ u ur die Spannungen, Verzerrungen und/oder Verschiebungen in Dickenrichtung einf€ u uhren, und es werden zu l€ o osende Gleichungen abgeleitet, die bez€ u uglich der Dimension reduziert sind (statt dreidimensionaler Feldgleichungen hat man ein System von zwei-oder eindimensionalen Feldgleichungen). € U Uber die Korrektheit und Genauigkeit dieses Konzepts wird berichtet.

A technique allowing inversion of the shale stiffness tensor from standard logging data: sonic velocities, density, porosity and clay content is developed. The inversion is based on the effective medium theory. The testing of the technique on laboratory measurements of the elastic wave velocities in shale samples shows that the inversion makes it possible to predict the elastic wave velocities V P , V S1 and V S2 in any direction within an error of a few per cent.

An extended version of the classical Generalized Backward Euler (GBE) algorithm is proposed for the numerical integration of a three-invariant isotropic-hardening elastoplastic model for cemented soils or weak rocks undergoing mechanical and non-mechanical degradation processes. The restriction to isotropy allows to formulate the return mapping algorithm in the space of principal elastic strains. In this way, an efficient and robust integration scheme is developed which can be applied to relatively complex yield surface and plastic potential functions. Moreover, the proposed algorithm can be linearized in closed form, thus allowing for quadratic convergence in the global Newton iteration. A series of numerical experiments are performed to illustrate the accuracy and convergence properties of the algorithm. Selected results from a finite element analysis of a circular footing on a soft rock layer undergoing chemical weathering are then presented to illustrate the algorithm performance at the boundary value problem level.

The Fresnel reflection coefficients R1, R2 for the planar interface between isotropic and generally bi-isotropic half spaces are studied in this communication. The functions (R1 + R2)/(1 + R1R2) and (R1-R2)/(1-R1R2) are seen to be simple functions of the optical material parameters of the media, as has been the case in the corresponding isotropic problem. Also pathological conditions, derived earlier elsewhere for reflection coefficients involving isotropic media, are generalized for the bi-isotropic (nonreciprocal chiral) case. Furthermore, the special cases of reflection from Pasteur or Tellegen media result from the analysis.

Many natural porous geological rock formations, as well as engineered porous structures, have fractal properties, i.e., they are self-similar over several length scales. While there have been many experimental and theoretical studies on how to quantify a fractal porous medium and on how to determine its fractal dimension, the numerical generation of a fractal pore structure with predefined statistical and scaling properties is somewhat scarcer. In the present paper a new numerical method for generating a three-dimensional porous medium with any desired probability density function (PDF) and autocorrelation function (ACF) is presented. The well-known Turning Bands Method (TBM) is modified to generate three-dimensional synthetic isotropic and anisotropic porous media with a Gaussian PDF and exponential-decay ACF. Porous media with other PDF's and ACF's are constructed with a nonlinear, iterative PDF and ACF transformation, whereby the arbitrary PDF is converted to an equivalent Gaussian PDF which is then simulated with the classical TBM. Employing a new method for the estimation of the surface area for a given porosity, the fractal dimensions of the surface area of the synthetic porous media generated in this way are then measured by classical fractal perimeter/area relationships. Different 3D porous media are simulated by varying the porosity and the correlation structure of the random field. The performance of the simulations is evaluated by checking the ensemble statistics, the mean, variance and ACF of the simulated random field. For a porous medium with Gaussian PDF, an average fractal dimension of approximately 2.76 is obtained which is in the range of values of actually measured fractal dimensions of molecular surfaces. For a porous medium with a non-Gaussian quadratic PDF the calculated fractal dimension appears to be consistently higher and averages 2.82. The results also show that the fractal dimension is neither strongly dependent of the porosity nor of the degree of anisotropy assumed.

The main aim of this study is to analyze fiber-reinforced polymer (FRP) bridge decks according to their material, cross-section, and shape geometry. Infill cell configurations of the decks (rectangular, triangular, trapezoidal, and honeycomb) were tested based on the FRP cell units available in the market. A comparison was made for each cell configuration in flat and curved bridge shapes. Another comparison was made between the material properties. Each model was computed for a composite layup material and a quasi-isotropic material. The quasi-isotropic material represents chopped fibers within a matrix. FE (finite element) analysis was performed on a total of 24 models using Abaqus software. The results show that the bridge shape geometry and infill configuration play an important role in increasing the stiffness, more so than improving the material properties. The arch shape of the bridge deck with quasi-isotropic material and chopped fibers was compared to the cross-ply laminate ...

In most existing pavement design methods, pavements are analyzed using linear elastic multilayer theory, with the assumption that isotropic conditions are present in the structural layers of the pavement section. There are however numerous researchers that disagree with this assumption and their research have proven that there are inherent and induced cross-anisotropic material behaviors in the structural layers of a pavement section. The problem facing the South African Mechanistic Design Method (SAMDM) is that this cross-anisotropic behavior of material may reduce the predicted life of pavement sections as described in the Technical Recommendations for Highway (TRH4). The objective of this study was to quantify the effect of cross-anisotropic conditions induced in the pavement layer on the predicted life based on the SAMDM approach.

The amplitudes and frequency content of seismic waves propagating through anisotropic formations may be strongly distorted by directionally dependent attenuation. Here, we analyze physical-modeling measurements of the P-wave attenuation coefficient in a transversely isotropic (TI) phenolic sample.

This paper investigates the effect of different conditions on the development of concrete expansions due to alkali-silica reaction (ASR), delayed ettringite formation (DEF) and their combination. The presence of products of the two reactions has been observed during structure diagnosis. The aim of this research is to study the two reactions in concretes with close mix designs but with various types of aggregate and moisture conditions. Measurements performed in the three directions of stress-free specimens showed that DEF expansions could be considered as isotropic for stress-free material. DEF expansions were largely influenced by the storage conditions (immersed in water or in sealed conditions). The volume of storage water modified the kinetics. Under sealed conditions, no expansions were measured for mortar containing non-reactive aggregate, while small positive strains were obtained for mortar containing reactive aggregate. In all cases, new water supply caused fast, large expansions. The different effects of alkali leaching and moisture conditions on DEF and ASR expansions are discussed.

Diffusion tensor magnetic resonance imaging (DT-MRI) provides a statistical estimate of a symmetric, second-order diffusion tensor of water, , in each voxel within an imaging volume. We propose a new normal distribution, () exp(1 2 : :), which describes the variability of in an ideal DT-MRI experiment. The scalar invariant, : : , is the contraction of a positive definite symmetric, fourth-order precision tensor, , and. A correspondence is established between : : and the elastic strain energy density function in continuum mechanics-specifically between and the second-order infinitesimal strain tensor, and between and the fourth-order tensor of elastic coefficients. We show that can be further classified according to different classical elastic symmetries (i.e., isotropy, transverse isotropy, orthotropy, planar symmetry, and anisotropy). When is an isotropic fourth-order tensor, we derive an explicit analytic expression for (), and for the distribution of the three eigenvalues of , (1 2 3), which are confirmed by Monte Carlo simulations. We show how can be estimated from either real or synthetic DT-MRI data for any given experimental design. Here we propose a new criterion for an optimal experimental design: that be an isotropic fourth-order tensor. This condition ensures that the statistical properties of (and quantities derived from it) are rotationally invariant. We also investigate the degree of isotropy of several DT-MRI experimental designs. Finally, we show that the univariate and multivariate distributions are special cases of the more general tensor-variate normal distribution, and suggest how to generalize () to treat normal random tensor variables that are of third-(or higher) order. We expect that this new distribution, (), should be useful in feature extraction; in developing a hypothesis testing framework for segmenting and classifying noisy, discrete tensor data; and in designing experiments to measure tensor quantities.

The dose incurred in an anisotropic environment depends on the orientation of the astronaut's body relative to the direction of the radiation field. The fluctuations in exposure of specific organs due to astronaut orientation are found to be a factor of 2 or more in a typical space habitation module and typical space radiations. An approximation function is found that

For a class of bounded reference configurations that are of partially rectangular type, the linear system of magneto-elasticity in two space dimensions is considered and a polynomial decay rate of the energy as time tends to infinity is proved.

1] We present a multiple step procedure for joint modeling of surface wave group velocity dispersion curves and teleseismic receiver functions for lithospheric velocity structure. The method relies on an initial grid search for a simple crustal structure, followed by a formal iterative inversion, an additional grid search for shear wave velocity in the mantle, and finally, forward modeling of transverse isotropy to resolve Love-Rayleigh surface wave dispersion discrepancy. It considers longer-period surface wave group velocity (SWGV) dispersion, allowing for the resolution of deeper structure compared to previous joint inversions. The grid search for simple crustal structure is facilitated using a library of precomputed receiver functions and SWGV dispersion curves. The iterative inversion improves fit to the data by increasing the number of layers in the crust when necessary. In order to fit the SWGV for periods greater than about 50 s, we perform a grid search over mantle velocities including the mantle lid and low-velocity zone, keeping the crustal structure fixed to the values from the previous step. In some cases a clear Love-Rayleigh discrepancy prevents a simultaneous fit of the group velocities with an isotropic model. The Love-Rayleigh discrepancy can be resolved by allowing shear wave transverse isotropy with a vertical symmetry axis (v SH À v SV differences) in the uppermost mantle. The method is applied to 10 stations in the Arabian Peninsula sampling various tectonic environments including active continental rifting and stable regions. The resulting shear velocity models confirm rapid crustal thinning of the Arabian Shield toward the Red Sea; however, we do not find strong evidence for crustal thickening toward the Arabian Platform. Our results suggest that the mantle lithosphere thickness varies regionally but that the mantle shear velocities beneath the Arabian Shield and Red Sea coast are generally anomalously low. Furthermore, our results indicate the presence of strong polarization anisotropy (up to about 10%) in the lithospheric upper mantle, in the vicinity of, as well as farther away from, the Red Sea. Our modeling yields v SV > v SH in the southwestern part of the Arabian Peninsula, consistent with vertical flow, and v SH > v SV in the northwestern part of the Arabian Peninsula and the continental interior, consistent with horizontal flow, indicating that the mantle flow pattern is not uniform along the axis of the Red Sea. (2006), A multistep approach for joint modeling of surface wave dispersion and teleseismic receiver functions: Implications for lithospheric structure of the Arabian Peninsula,

The present paper concerns the formulation and the evolution of the non symmetrical growth of an avascular cancerous cell colony in an analytical mathematical fashion. Although most of the existing research considers spherical tumours, here we work in the frame of a more general case of the prolate spheroidal geometry. The tumour lies inside a host spheroidal shell which provides vital nutrients, receives the debris of the dead cells and also transmittes to the tumour the pressure imposed by the surrounding on its exterior boundary. Under the aim of studying the evolution of the exterior tumour boundary, we focus on the exterior conditions under which such a geometrical structure can be sustained. To that purpose, the corresponding nutrient concentration, the inhibitor concentration and the pressure field are calculated analytically providing the necessary data for the evolution equation to be solvable. It turns out that an avascular tumour can exhibit a prolate spheroidal growth on...

This paper presents some fundamental concepts of seismic anisotropy specifically those which have hexagonal symmetry (ordinarily called transversely isotropic in geophysics jargon). There were made calculations of phase and group angles at a planar interface separating an anisotropic media, using routines written in Matlab ® and Maple ® language. The slowness surfaces of the qP, qSV and SH wave, as well as the ray paths in these two media were also estimated. Although only the simplest situations are discussed, this paper is useful as a first step in understanding the fractured media, because it contains examples, software routines, and a reviewing of the basic concepts and formulas of wave propagation.

1Technology Cooperation Agreement: Ecopetrol SA - Instituto Colombiano del Petróleo, AA 4185 Bucaramanga, Santander, Colombia; Universidad Industrial de Santander (UIS) -Petroseismic Group, Bucaramanga, Santander, Colombia 2Geophysical Group, Research Unit, ...

We characterize the angular polyspectra, of arbitrary order, associated with isotropic fields defined on the sphere S 2 = (x, y, z) : x 2 + y 2 + z 2 = 1 . Our techniques rely heavily on group representation theory, and specifically on the properties of Wigner matrices and Clebsch-Gordan coefficients. The findings of the present paper constitute a basis upon which one can build formal procedures for the statistical analysis and the probabilistic modelization of the Cosmic Microwave Background radiation, which is currently a crucial topic of investigation in cosmology. We also outline an application to random data compression and "simulation" of Clebsch-Gordan coefficients.

In this paper, two complementary methods are introduced to analyze the sensitivity of a three-degree-of-freedom (3-DOF) translational parallel kinematic machine (PKM) with orthogonal linear joints: the Orthoglide. Although these methods are applied to a particular PKM, they can be readily applied to 3-DOF Delta-Linear PKM such as ones with their linear joints parallel instead of orthogonal. On the one hand, a linkage kinematic analysis method is proposed to have a rough idea of the influence of the length variations of the manipulator on the location of its end-effector. On the other hand, a differential vector method is used to study the influence of the length and angular variations in the parts of the manipulator on the position and orientation of its end-effector. Besides, this method takes into account the variations in the parallelograms. It turns out that variations in the design parameters of the same type from one leg to another have the same effect on the position of the end-effector. Moreover, the sensitivity of its pose to geometric variations is a minimum in the kinematic isotropic configuration of the manipulator. On the contrary, this sensitivity approaches its maximum close to the kinematic singular configurations of the manipulator.

We review recent progresses on anomalous scaling and universality in anisotropic and homogeneous hydrodynamic turbulent flows. As a central matter, we discuss the validity and the limits of classical ideas of statistical isotropy restoration. Finally, we comment on a still open issue, the observed different scaling behaviour of longitudinal and transverse velocity increment moments in purely statistically isotropic ensemble.

1Technology Cooperation Agreement: Ecopetrol SA - Instituto Colombiano del Petróleo, AA 4185 Bucaramanga, Santander, Colombia; Universidad Industrial de Santander (UIS) -Petroseismic Group, Bucaramanga, Santander, Colombia 2Geophysical Group, Research Unit, ...

The fabric of polar ice exhibiting orthotropic symmetry (i.e., three orthogonal planes of refiexional symmetry) is described by a continuous orientation distribution function (ODF) which represents the density of grains whose c axes have a given orientation. For a given ODF, the constitutive law of such anisotropic ice is obtained by a. homogenization procedure based on the assumptions that the ice single crystM beha. ves as • linear transversely isotropic m•teria,1 •nd that the grains experience a uniform state of stress. Under loading conditions which respect the symmetries of orthotropy, the evolution of the macroscopic behavior of a polycrystal of ice, a,s well a.s the evolution of its fabric, are fully determined by analytical expressions. The solution for a transversely isotropic polycrystal is deduced as a reduction from the orthotropic case. In this special configuration, the results from the present model are compa. red to fabric measurements made by Thorsteinsson et al. [1997] on the GRIP ice core. Polycrystalline polar ice deforms mainly by intracrystalline dislocation glide [Pimienta et al., 1987]. Under creep conditions, the minimum strain rate i and the deviatoric stress s are related by a Norton-Hoff type 17,797 17,798 GAGLIARDINI AND MEYSSONNIER: A MODEL FOR ORTHOTROPIC ICE power law, known as Glen's law in glaciology: 2 •-B. r"-xs, (2) where r is the second invariant, of s and B• is a temperature-dependent parameter. For stresses greater than 0.1 MPa, the exponent n is 3 [Duval et al., 1983]. The different value of n as compared to the single crystal suggests that the polycrystalline deformation involves mechanisms other than basal glide to resolve strain incompatibilities that, would arise between adjacent, grains. Such mechanisms probably include dislocation climb [Duval et al., 1983]. For stresses less than 0.1 MPa, which is a more appropriate range for polar ice as the order of magnitude of the deviatoric stress in an ice sheet is less than 0.05 MPa [Lliboutw and Duval, 1985], there are some indications of a stress exponent less than 2, and possibly close to 1, from laboratory tests and field data analysis [Doake and Wolff, 1985;

T his paper presents some fundamental concepts of seismic anisotropy specifically those which have hexagonal symmetry (ordinarily called transversely isotropic in geophysics jargon). There were made calculations of phase and group angles at a planar interface separating an anisotropic media, using routines written in Matlab ® and Maple ® language. The slowness surfaces of the qP, qSV and SH wave, as well as the ray paths in these two media were also estimated. Although only the simplest situations are discussed, this paper is useful as a first step in understanding the fractured media, because it contains examples, software routines, and a reviewing of the basic concepts and formulas of wave propagation.

We prove that the innermost wavefront-slowness sheet of a Hookean solid is convex, whether or not it is detached from the other sheets. This theorem is valid for the generally anisotropic case, and it is an extenstion of theorems whose proofs require the detachment of the innermost sheet. Even though the Hookean solids that represent most materials encountered in seismology exhibit a detached innermost sheet, the positive definiteness of the elasticity tensor, which is its sole fundamental constraint, allows for the existence of both detached and nondetached cases. Besides the foundational considerations, the omnipresence of computer methods requires that we investigate cases that -even if not commonly encountered -are within the realm of physical possibilities, and can appear as the output of modelling. We exemplify the theorem -proven for a general Hookean solid -using a particular case of transverse isotropy. For that case, we show that the innermost sheet exhibits a

A nonholonomic system subjected to external noise from the environment, or internal noise in its own actuators, will evolve in a stochastic manner described by an ensemble of trajectories. This ensemble of trajectories is equivalent to the solution of a Fokker-Planck equation that typically evolves on a Lie group. If the most likely state of such a system is to be estimated, and plans for subsequent motions from the current state are to be made so as to move the system to a desired state with high probability, then modeling how the probability density of the system evolves is critical. Methods for solving Fokker-Planck equations that evolve on Lie groups then become important. Such equations can be solved using the operational properties of group Fourier transforms in which irreducible unitary representation (IUR) matrices play a critical role. Therefore, we develop a simple approach for the numerical approximation of all the IUR matrices for two of the groups of most interest in robotics: the rotation group in three-dimensional space, SO(3), and the Euclidean motion group of the plane, SE(2). This approach uses the exponential mapping from the Lie algebras of these groups, and takes advantage of the sparse nature of the Lie algebra representation matrices. Other techniques for density estimation on groups are also explored. The computed densities are applied in the context of probabilistic path planning for kinematic cart in the plane and flexible needle steering in three-dimensional space. In these examples the injection of artificial noise into the computational models (rather than noise in the actual physical systems) serves as a tool to search the configuration spaces and plan paths. Finally, we illustrate how density estimation problems arise in the characterization of physical noise in orientational sensors such as gyroscopes.

The variogram provides an important method for measuring the dependence of attribute values between spatial locations. Suppose that the nature of the sampling process leads to the presence of clustered data; it would be advisable to use a variogram estimator that aims to adjust for clustering of samples. In this setting, the use of a non-parametric weighted estimator, obtained by considering an inverse weight to a given neighbourhood density combined with the kernel method, seems to have a satisfactory behaviour in practice. This paper pursues a theoretical study of the cluster robust estimator, by proving that it is asymptotically unbiased as well as consistent and by providing criteria for selection of the bandwidth parameter and the neighbourhood radius. Numerical studies are also included to illustrate the performance of the considered estimator and the suggested approaches.

Solutions to the two, coupled, ordinary nonlinear differential equations for a semi-infinite circular elastic tube subjected to edge loads and undergoing small axisymmetric strains, but arbitrarily large axisymmetric rotations-the simplified Reissner equations-are analyzed. First, with the aid of a Green's function, the differential equations and boundary conditions are transformed to a complex-valued integral equation. From this equation existence, uniqueness, boundedness, and rate of decay are extracted for a dimensionless stress function, /, and an angle of rotation, /?, for sufficiently small edge data. It is shown that these unique solutions must decay at least as fast as the linear solution. Second, it is shown that any solution that decays to zero must, at a sufficiently large distance from the edge, decay at the linear rate. Third, rates of decay are established for any solution for which the or the sup norm of /? has certain bounds. Finally, an energy (or Lyapunov) function E, defined on solutions of the differential equations, is constructed and under certain a priori restrictions on the angle of rotation, three different upper bounds on E are obtained. These also provide exponential decay estimates for solutions. The energy approach is examined with a view to more general shells where a Green's function may not be readily available. 1. Introduction. We shall be concerned with a homogeneous, elastically isotropic, semi-infinite circular tube (cylindrical shell) of constant thickness h and midsurface radius R under axisymmetric edge loads. In classical, (first-approximation) theory, the linear field equations can be reduced to the following coupled system of secondorder ordinary differential equations [1]: f"-e2 f + /? = 0, p"-e2p-f = 0, 0 < x < oc. (1.1) Here, / is a dimensionless stress function, /? is the angle of rotation of a generator

High-pressure isotropic compression tests were carried out on samples of a uniform fine sand at different initial specific volumes, reinforced with randomly distributed polypropylene fibers ͑0.5% by weight, 24 mm in length and 0.023 mm thick͒, as well as on a nonreinforced sand specimen. Analysis of the results identified changes in the isotropic compression behavior due to the random inclusion of fibers into the sand. Two distinct and parallel normal compression lines were observed for the fiber-reinforced and nonreinforced sand. The fibers were exhumed after testing and it was found that fibers had both extended and broken, indicating that the fibers tend to suffer large plastic tensile deformations before breaking and that the fibers act under tension even in isotropic compression.

In this work, a new concrete model that strongly couples continuum-damage-mechanics to elastoplasticity is presented. The model incorporates a plasticity yield criterion written in terms of the nominal/damaged, rather than effective, stress space. Tensile and compressive behaviors are modelled through the damage-affected-multi-hardening nature of the plasticity yield criterion and the introduction of two (tensile and compressive) plasticity-affected isotropic damage initiation/growth criteria, where plasticity variables are added to the definition of the tensile and compressive growth functions of damage. A non-associative plastic flow rule is used to control inelastic dilatancy. Specific expressions of the elastic/damage and plastic/damage components of the Helmholtz free energy function are presented. The constitutive equations of the model are consistently derived within a sound framework of irreversible thermodynamics. These free energy expressions are used to define (one-plasticity) and (two-damage) interconnected dissipation mechanisms. The elastic-plastic-damage, elastic-plastic, and elastic-damage tangent operators are also derived. A computational algorithm for the numerical integration of the

Diffusion Tensor MRI (DT-MRI) provides a statistical estimate of a symmetric 2 nd -order diffusion tensor, D, for each voxel within an imaging volume. We propose a new normal distribution, p(D) ~ exp(-1/2 D:A:D), for a tensor random variable, D. The scalar invariant, D:A:D, is the contraction of a positive definite symmetric 4 th -order precision tensor, A, and D. A formal correspondence is established between D:A:D and the elastic strain energy density function in continuum mechanics.

Anisotropic attenuation can provide sensitive attributes for fracture detection and lithology discrimination. This paper analyzes measurements of the P-wave attenuation coefficient in a transversely isotropic sample made of phenolic material. Using the spectral-ratio method, we estimate the group ͑effective͒ attenuation coefficient of P-waves transmitted through the sample for a wide range of propagation angles ͑from 0°to 90°͒ with the symmetry axis. Correction for the difference between the group and phase angles and for the angular velocity variation help us to obtain the normalized phase attenuation coefficient A governed by the Thomsenstyle attenuation-anisotropy parameters ⑀ Q and ␦ Q . Whereas the symmetry axis of the angle-dependent coefficient A practically coincides with that of the velocity function, the magnitude of the attenuation anisotropy far exceeds that of the velocity anisotropy. The quality factor Q increases more than tenfold from the symmetry axis ͑slow direction͒ to the isotropy plane ͑fast direction͒. Inversion of the coefficient A using the Christoffel equation yields large negative values of the parameters ⑀ Q and ␦ Q . The robustness of our results critically depends on several factors, such as the availability of an accurate anisotropic velocity model and adequacy of the homogeneous concept of wave propagation, as well as the choice of the frequency band. The methodology discussed here can be extended to field measurements of anisotropic attenuation needed for AVO ͑amplitude-variation-with-offset͒ analysis, amplitude-preserving migration, and seismic fracture detection.

Now a days there are few areas on the surface of earth having a relatively constant BSC and can serving as radar calibration targets. Few Examples include the Amazon rain forest, the Greenland ice sheet etc. In this paper we analyse a consistent set of Ku-band radar measurements from the QuickSCAT data mission for about ten years. To study the long-term variation of observed BSC, a set of data is required from Earth's surface this band. A global survey of potential constant land targets is performed. To identify the locations of the best natural calibration targets Quantitative measurements of temporal and spatial variabilities (homogeneity) and isotropy are used. By concentrating on low variation areas, useful calibration targets for future radar missions and for inter calibration between existing radars can be identified. By focusing on regions with little spatial heterogeneity, the temporal variations on diurnal, seasonal, and also decadal scales in homogenous natural terrain types including rain forest, dry brushy areas, and ice sheets can be analysed. The Greenland ice sheet shows a significant trend in backscatter in recent years, and therefore, may not be a suitable calibration site anymore. Another location for a good stable target is a dry brushy area in the Sahara, which shows comparable stability and isotropy with those of the Amazon, Congo, and Antarctica.

hree-dimensional laser Doppler anemometry measurements have been performed in a vessel stirred by a Rushton turbine. The mean flow field, Reynolds stresses and triple velocity correlations have been determined throughout the entire vessel. The anisotropy of turbulence has been assessed through the analysis of the invariants of the anisotropy. The results indicated that turbulence deviates significantly from the isotropic state not only in the impeller region but also near the bottom of the vessel. The triple velocity correlations measured indicated that the turbulent diffusion of the kinetic energy may be significant over only one-quarter of the circulation loops.

We revisit the singular eigensolution to the steady state one-speed transport equation for an isotropically scattering and multiplying heterogeneous slab. It is proved that this solution is a sum of Stieltjes integrals over the resolvent set of only the operator of multiplication by the angular variable.

The strengths of geomaterials and their variation under different factors are investigated in this paper. First, a general isotropic variation of a strength criterion is proposed for describing the critical state and peak strengths of geomaterials. Second, the proposed criterion is extended to describe the effect of anisotropy on the peak strength. After an analysis of experimental data, the hypothesis is made that the failure of an element of geomaterial generally occurs in a particular plane when the applied shear stress in that plane reaches the shear resistance of the material. Therefore, the variation of the peak strength of anisotropic materials should be described in terms of the stress tensor applied, a vector parameter defining the position of the potential failure plane of the material, and the material properties. A general failure criterion for geomaterials with cross anisotropy is obtained then from the proposed isotropic strength criterion. The proposed criterion is demonstrated to well represent both the isotropic and anisotropic strengths of various geomaterials. Finally, a general anisotropic criterion is introduced.

We report measurements of long-period P wave polarization (P pol) in Australia and Tahiti made by combining modeling of the polarization deviation and harmonic analysis. The analysis of the deviation of the horizontal polarization of the P wave as a function of event back azimuth may be used to obtain information about (1) sensor misorientation, (2) dipping discontinuities, (3) seismic anisotropy, and (4) velocity heterogeneities beneath a seismic station. The results from harmonic analysis and a grid search using Snell's law suggest the presence of a dipping seismic discontinuity beneath stations CTAO and CAN in Australia. These results are consistent with published receiver function studies for these stations. The P pol fast axis orientation is close to the N-S absolute plate motion direction at station TAU (Tasmania), which may be due to plate-motion-driven alignment of olivine crystals in the asthenosphere. Interestingly, measurements of SKS splitting at Tahiti (French Polynesia) show an apparent isotropy, whereas an inversion of P pol observations at PPTL seismic station located in Tahiti suggests the presence of two anisotropic layers. The fast axis azimuth is oriented E-W in the upper layer, and it is close to the NW-SE orientation in the lower layer. Since P pol orientations are used for real-time earthquake locations, especially in poorly instrumented areas such as the South Pacific, we show that the bias from anisotropy and sensor misorientation determined here can be corrected to improve the location accuracy, which yields fundamental data for rapid location necessary for effective tsunami warning.

The broadband recording site at Warramunga (WRA) in the Northern Territory of Australia provides good coverage of seismic wave propagation through the upper mantle for sources in the earthquake belt through Indonesia and New Guinea. $ waves recorded on the radial ($V) and tangential (SH) components are of comparable quality because the hard-rock recording site minimizes the influence of coupling to P on the radial component. Refracted $ waves from the uppermantle transition zone show a clear advance of $H wave arrivals compared with $V. Eleven polarization analyses of waves returned from the transition zone yield an average time shift of 2.3 s with the fast direction scattered about the transverse direction. Nine polarization measurements of waves returned from the top of the lower mantle yield an average time shift of 1.7 s, again with the fast direction near the transverse. No appreciable time differences are observed between the radial and transverse polarizations for paths refracted within the lithospheric lid. Because the observations of shear wave splitting in waves passing through the low-velocity zone, '•he transition zone, and the top of the lower mantle are not coherent in their absolute polarization, the cause cannot lie in azimuthal anisotropy at shallow depths under the WRA station. The most plausible exph•nation is transverse isotropy in shear within the low-velocity zone under the unusually thick mantle "lid" under Australia. A possible contribution may come from anisotrol•y in •-olivine at the top of the upper mantle transition zone. Transverse isotropy in the 200-kin-thick layer below the lithosphere down to the transition zone with a 1% faster shear wave speeds for horizontal polarization compared with vertical polarization will explain the splitting data. For this asthenospheric region the level of anisotropy is quite reasonable and the polarization is consistent with lateral flow. The geometry of the available paths for waves propagating within the mmltle lid i• not sufficient to place constraints on the anisotropic properties of this heterogeneous a,nd low-loss region.

T hree-dimensional laser Doppler anemometry measurements have been performed in a vessel stirred by a Rushton turbine. The mean flow field, Reynolds stresses and triple velocity correlations have been determined throughout the entire vessel. The anisotropy of turbulence has been assessed through the analysis of the invariants of the anisotropy. The results indicated that turbulence deviates significantly from the isotropic state not only in the impeller region but also near the bottom of the vessel. The triple velocity correlations measured indicated that the turbulent diffusion of the kinetic energy may be significant over only one-quarter of the circulation loops.

1] We report measurements of long-period P wave polarization (P pol ) in Australia and Tahiti made by combining modeling of the polarization deviation and harmonic analysis. The analysis of the deviation of the horizontal polarization of the P wave as a function of event back azimuth may be used to obtain information about (1) sensor misorientation, (2) dipping discontinuities, (3) seismic anisotropy, and (4) velocity heterogeneities beneath a seismic station. The results from harmonic analysis and a grid search using Snell's law suggest the presence of a dipping seismic discontinuity beneath stations CTAO and CAN in Australia. These results are consistent with published receiver function studies for these stations. The P pol fast axis orientation is close to the N-S absolute plate motion direction at station TAU (Tasmania), which may be due to plate-motion-driven alignment of olivine crystals in the asthenosphere. Interestingly, measurements of SKS splitting at Tahiti (French Polynesia) show an apparent isotropy, whereas an inversion of P pol observations at PPTL seismic station located in Tahiti suggests the presence of two anisotropic layers. The fast axis azimuth is oriented E-W in the upper layer, and it is close to the NW-SE orientation in the lower layer. Since P pol orientations are used for real-time earthquake locations, especially in poorly instrumented areas such as the South Pacific, we show that the bias from anisotropy and sensor misorientation determined here can be corrected to improve the location accuracy, which yields fundamental data for rapid location necessary for effective tsunami warning.

Results of molecular-mechanics simulations of axial and torsional deformations of a single wall carbon nanotube are used to find Young's modulus, the shear modulus, and the wall thickness of an equivalent continuum tube made of a linear elastic isotropic material. These values are used to compare the response of the continuum tube in bending and buckling with that obtained from the molecular mechanics simulations. It is found that the strain energy of bending deformation computed from the Euler-Bernoulli beam theory matches well with that obtained from the molecular-mechanics simulations. The molecular-mechanics predictions of the critical strains for axial buckling and shell wall buckling do not match well with those derived from the Euler buckling formula and the Donnell shell theory.